Fachbereich
Mathematik und Statistik

Universität
Konstanz

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Block Seminar on Profinite Groups

This is the homepage of the Block Seminar on Profinite Groups and Infinite Galois Theory.

Organisers

Prof. Dr. Salma Kuhlmann and Michele Serra

Schedule

All talks will take place in F426 from 10:00 to 11:30, according to the table below.

DateSpeakerTopic
Mo06.03MicheleIntroduction I
Mi08.03MicheleIntroduction II
Mi15.03MicheleIntroduction III, profinite and p-adic integers
Fr17.03MarcoBasic properties of profinite groups
Mo20.03ZhelunSylow theorems for profinite groups
Mi22.03MichaelGenerators
Mo27.03PiaProfinite groups as Galois groups
Mi29.03FabianProcyclic groups

Here you can find

  • Outlines of the topics covered;
  • Notes of the first three talks;
  • Students' evaluation of the course.

    • Contents

    The goal of these lectures is to give an introduction to the theory of profnite groups. These are topological groups obtained as projective limits of finite groups. These groups are of great relevance in many domains: Number Theory (the p-adic integers are an example of a profnite group), Algebraic Geometry (fundamental groups considered there are profnite) and Galois Theory (the Galois groups of infinite Galois extensions are also profnite). We will start by introducing inverse systems and general topological groups. Then we will define profnite groups and study their main properties, and use them in the study of infinite Galois extensions of fields, aiming to generalise the fundamental theorem of Galois Theory to such extensions.

    Rules to get credits

    Active participation is required in all the talks, unless a valid reason prevents you from doing so. All participants will have to speak once, those taking the Seminar as Fachseminar (4ECTS) are additionally asked to produce a report of their talk.

    Prerequisites

    Only basic knowledge of Algebra (group theory, field theory and Galois theory) and point-set topology is required.

    Personal Tutorial

    A personal tutorial of two hours per week will be offered to all the participants to discuss problems or questions about the topics covered and advise on the preparation of their talks. The tutorial will normally take place on Fridays, 10:00-11:30, with the exception of March the 17th when there will be a talk. On that week the tutorial will be scheduled on appointment.

    References

  • L. Ribes and P. Zalesskii, Profinite groups, A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 2010, ISBN: 978-3-642-01641-7
  • J.S. Wilson, Profinite Groups, LMS Monographs, Oxford Science Publications, 1998, ISBN: 0-19-850082-3
  • J. Milne, Fields and Galois Theory, Notes available online: link
  • B. Osserman, Infinite Galois Theory, Notes available online: link
  • Further references on specific topics might be given in due time

    • Target group: BA (having completed 3rd semester), MA, LA.
    Credit: 3 ECTS as Proseminar / 4 ECTS as Fachseminar
    Language: English/German